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# Trigonometry: Polar and Rectangular Coordinates

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Convert the equation to polar coordinates and simplify.
1. a. xy = 1
b. (x^2 + y62)^2 = 2xy

Express the equation in rectangular coordinates.
2. a. r = 3sin(t)
b. r = 4cos3(t)
r^2 = 4sin2(t)
r = 4/(2 + cos(t))

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https://brainmass.com/math/trigonometry/trigonometry-polar-rectangular-coordinates-513319

## SOLUTION This solution is FREE courtesy of BrainMass!

1. First x=rcos(t), y=rsin(t).
a. Xy=1.
(rcos(t))(rsin(t))=1
r^2(cos(t)sin(t)=1
r^2sin(2t)=2.
Therefore, the polar equation is r^2sin(2t)=2.

b. Since x^2+y^2=r^2
(r^2)^2=2r^2cos(t)sin(t),
We first cancel out r^2: r^2=sin(2t).
Therefore, the polar equation is r^2=sin(2t).

2. X=rcos(t), y=rsin(t).
a. r=3sin(t)
we multiple r on both sides:
r^2=3rsin(t)
x^2+y^2=3y
therefore, the rectangular equation is x^2-3y+y^2=0
b. r=4cos(3t)
cos(3t)=cos(2t)cos(t)-sin(t)sin(2t)=(2cos^2(t)-1)cos(t)-2sin^2(t)cos(t)=2cos^3(t)-cos(t)-2cos(t)(1-cos^2(t))=4cos^3(t)-3cos(t).

r=4(4cos^3(t)-3cos(t))=16cos^3(t)-12cos(t)
multiply r^3 on both sides:
r^4=16(rcos(t))^3-12r^2(rcos(t)
(x^2+y^2)^2=16x^3-12(x^2+y^2)x
X^4+2x^2y^2+y^4=4x^3-12xy^2
Therefore, the rectangular equation is
x^4+2x^2y^2+y^4-4x^3+12xy^2=0.

c. r^2=4sin(2t)
r^2=4*2sin(t)cos(t)
multiple r^2 on both sides:
r^4=8(rsin(t))(rcos(t)
(x^2+y^2)^2=8xy
Therefore, the rectangular equation is
(x^2+y^2)^2=8xy.

d. r=4/(2+cos(t)
r(2+cos(t))=4
2r=(4-rcos(t))
4r^2=(4-rcos(t))^2
4(x^2+y^2)=(4-x)^2
4x^2+4y^2=x^2-8x+16
3x^2+8x+4y^2-16=0.
Therefore, the rectangular equation is
3x^2+8x+4y^2-16=0.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

© BrainMass Inc. brainmass.com September 27, 2022, 4:17 pm ad1c9bdddf>
https://brainmass.com/math/trigonometry/trigonometry-polar-rectangular-coordinates-513319